Approaching a quantum critical point (QCP) has been an effective route to stabilize superconductivity. While the role of magnetic QCPs has been extensively discussed, similar exploration of a structural QCP is scarce due to the lack of suitable systems with a continuous structural transition that can be conveniently tuned to 0~K. Using inelastic X-ray scattering, we examine the phonon spectrum of the nonmagnetic quasi-skutterudite (CaSr)$3$Rh$4$Sn$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$, which represents a precious system to explore the interplay between structural instabilities and superconductivity by tuning the Ca concentration $x$. We unambiguously detect the softening of phonon modes around the M point on cooling towards the structural transition. Intriguingly, at $x=0.85$, the soft mode energy squared at the M point extrapolates to zero at $(-5.7\pm 7.7)$~K, providing the first compelling microscopic evidence of a structural QCP in (CaSr)$3$Rh$4$Sn$\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$. The enhanced phonon density-of-states at low energy provides the essential ingredient for realizing strong-coupling superconductivity near the structural QCP.